The Distance and Midpoint Formulas in Trigonometry

Introduction

The distance and midpoint formulas are essential tools in mathematics, providing a way to calculate the distance between two points and the point that lies exactly halfway between them in a Cartesian plane. These concepts are foundational for understanding geometric relationships in trigonometry and coordinate geometry.

The Distance Formula

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

This formula is derived from the Pythagorean theorem, where the difference in $x$ -coordinates and $y$ -coordinates forms the legs of a right triangle, and the distance is the hypotenuse.

The Midpoint Formula

The midpoint of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

This formula finds the average of the $x$ -coordinates and $y$ -coordinates, giving the coordinates of the midpoint.

Example: Distance and Midpoint

Find the distance and midpoint of the points $(2, 3)$ and $(6, 7)$ .

Step 1: Calculate the Distance

Use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substitute $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (6, 7)$ :

$d = \sqrt{(6 - 2)^2 + (7 - 3)^2}$

$d = \sqrt{4^2 + 4^2}$

$d = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}$

The distance is $4\sqrt{2}$ units.

Step 2: Calculate the Midpoint

Use the midpoint formula:

$\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Substitute $(x_1, y_1) = (2, 3)$ and $(x_2, y_2) = (6, 7)$ :

$\text{Midpoint} = \left(\frac{2 + 6}{2}, \frac{3 + 7}{2}\right)$

$\text{Midpoint} = \left(4, 5\right)$

The midpoint is $(4, 5)$ .

Practice Problems

Find the distance and midpoint of the points $(1, 2)$ and $(5, 6)$ .
Determine the distance and midpoint of $(3, 4)$ and $(-1, -2)$ .
Prove that the midpoint of $(0, 0)$ and $(8, 10)$ lies on the line segment connecting them.